Research Article Estimation of Cyclic Interstory Drift

Research ArticleEstimation of Cyclic Interstory Drift Capacity of Steel FramedStructures and Future Applications for Seismic Design

Edén Bojórquez,1 Alfredo Reyes-Salazar,1 Sonia E. Ruiz,2 and Amador Terán-Gilmore3

1 Facultad de Ingenierıa, Universidad Autonoma de Sinaloa, Calzada de las Americas y Boulevard Universitarios S/N,Ciudad Universitaria, 80040 Culiacan Rosales, SIN, Mexico

2 Coordinacion de Mecanica Aplicada, Instituto de Ingenierıa, Universidad Nacional Autonoma de Mexico,Ciudad Universitaria, 04510 Coyoacan, Mexico

3 Departamento de Materiales, Universidad Autonoma Metropolitana, Avenida San Pablo 180, 02200 Mexico, DF, Mexico

Correspondence should be addressed to Eden Bojorquez; [email protected]

Received 8 February 2014; Revised 26 April 2014; Accepted 31 May 2014; Published 26 June 2014

Academic Editor: Chung-Che Chou

Copyright © 2014 Eden Bojorquez et al.This is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Several studies have been devoted to calibrate damage indices for steel and reinforced concrete members with the purpose ofovercoming some of the shortcomings of the parameters currently used during seismic design. Nevertheless, there is a challengeto study and calibrate the use of such indices for the practical structural evaluation of complex structures. In this paper, anenergy-based damage model for multidegree-of-freedom (MDOF) steel framed structures that accounts explicitly for the effectsof cumulative plastic deformation demands is used to estimate the cyclic drift capacity of steel structures. To achieve this, seismichazard curves are used to discuss the limitations of the maximum interstory drift demand as a performance parameter to achieveadequate damage control. Then the concept of cyclic drift capacity, which incorporates information of the influence of cumulativeplastic deformation demands, is introduced as an alternative for future applications of seismic design of structures subjected to longduration ground motions.

1. Introduction

Currently, the maximum interstory drift and ductilitydemands are targeted as response parameters to achieve ade-quate structural performance of earthquake-resistant struc-tures. Nevertheless, the use of these parameters is not com-pletely justified for buildings subjected to long durationground motions. In fact, ample evidence suggests that thestructural performance of buildings subjected to long dura-tion groundmotions is not adequately characterized throughmaximum deformation demands [1]. Therefore, in somecases, the effect of cumulative plastic deformation demandsmust be explicitly considered.

Although different energy-based methodologies that aimat providing earthquake-resistant structures with adequateenergy dissipating capacity have been proposed [2–5], cur-rently, the most popular response parameter worldwidefor seismic design of buildings is the maximum interstory

drift. Although the use of energy concepts during practicalearthquake-resistant design requires further developments,the influence of cumulative demands should be incorpo-rated into seismic design of structures subjected to longduration motions. A viable manner to account explicitlyfor cumulative plastic deformation demands is the use ofa cyclic (reduced) drift capacity, which in concept is sim-ilar to the target ductility concept formulated by [6]. Theaim of this paper is firstly to, introduce an energy-baseddamage index which explicitly accounts for the effects ofcumulative plastic deformation demands in steel frames,secondly, to compare demand hazard curves obtained formoment-resisting steel frames in terms of the energy-based damage index and the maximum interstory drift,and finally, to provide for the steel frames drift capacitythresholds (denoted cyclic drift capacity) that account forcumulative damage and that yield adequate levels of reliabil-ity.

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 496206, 9 pageshttp://dx.doi.org/10.1155/2014/496206

2 The Scientific World Journal

2. Energy-Based Damage Index for SteelFramed Structures

Energy-basedmethodologies are focused on providing struc-tures with energy dissipating capacities that are equal toor larger than their expected energy demands [2, 7]. Thedesign requirement for an earthquake-resistant structure canbe formulated in these terms as

Energy Capacity ≥ Energy Demand. (1)

Among all the energies absorbed and dissipated by astructure, the plastic dissipated hysteretic energy, 𝐸

𝐻, is

clearly related to structural damage. 𝐸𝐻

can physically beinterpreted by considering that it is equal to the total areaunder all the hysteresis loops that a structure undergoesduring a groundmotion.Therefore, it is convenient to express(1) in terms of plastic dissipated hysteretic energy:

𝐸HC ≥ 𝐸HD, (2)

where 𝐸HC is the plastic hysteretic energy capacity and 𝐸HDis its corresponding energy demand. Equation (2) can bereformulated as an energy-based damage index:

𝐼DE =𝐸HD𝐸HC

≤ 1. (3)

Within the context of (3), the performance level orcondition that implies that the energy demand on the systemis equal to its corresponding capacity will be considered asthe failure of that system. Hence, while 𝐼DE equal to onecorresponds to failure of the structural system, a value ofzero implies no structural damage (elastic behavior impliesno structural damage). From a physical point of view, thisequation represents a balance between the structural capacityand demand in terms of energy. In this sense, this formulationfollows the direction initially established by Housner in 1956for an energy-based design. According to (3), structuraldamage depends on the balance between the plastic hystereticenergy capacity and demand on the structure. While theplastic hysteretic energy demand can be obtained throughdynamic analysis, a challenge exists to define the plastichysteretic energy capacity of a structure. Nevertheless, flex-ural plastic behavior is usually concentrated at the ends ofthe structural members that make up a frame and in theparticular case of 𝑊 steel shapes in the flanges. The plastichysteretic energy capacity of a steel member that forms partof a structural frame can be estimated as follows [2]:

𝐸HCm = 2𝑍𝑓𝑓𝑦𝜃𝑝𝑎, (4)

where 𝑍𝑓is the section modulus of the flanges, 𝑓

𝑦is the

yield stress, and 𝜃𝑝𝑎

is its cumulative plastic rotation capacity.Note that the above equation considers that plastic energyis dissipated exclusively through plastic behavior at bothends of a steel member. Equation (4) can be used togetherwith (3) to evaluate the level of structural damage in steelmembers. However, it is convenient to normalize 𝐸

𝐻for

damage evaluation purposes [8, 9]:

𝐸𝑁=𝐸𝐻

𝐹𝑦𝛿𝑦

, (5)

where 𝐹𝑦and 𝛿

𝑦are the strength and displacement at first

yield, respectively. Equation (3) can be expressed in terms of𝐸𝑁as follows:

𝐼DEN =𝐸ND𝐸NC

≤ 1, (6)

where the parameters involved in (6) have similar meaningsas those used in (3). The advantage of formulating theproblem in terms of 𝐸

𝑁is that this is a more stable parameter

and thus can be used in quantitative terms for practicalpurposes. The energy-based damage index proposed hereincorresponds to the ratio between the normalized hystereticenergy demand and normalized hysteretic energy capacity,and the condition of failure is assumed to be 𝐼DEN equalto one. In the case of MDOF steel structures, the principalchallenge for the practical use of (6) is the definition ofthe energy capacity of the structure in terms of that ofits structural members. Through the consideration that inregular steel frames the energy is dissipated exclusively by thebeams (which is a reasonable assumption for strong column-weak beam structural systems), their energy capacity can beestimated through a modified version of (4) [5]:

𝐸NC =∑𝑁𝑆

𝑖=1(2𝑁𝐵𝑍𝑓𝐹𝑦𝜃𝑝𝑎𝐹EH𝑖)

𝐶𝑦𝐷𝑦𝑊

, (7)

where 𝑁𝑆and 𝑁

𝐵are the number of stories and bays in the

building, respectively, 𝐹EH𝑖 is an energy participation factorthat accounts for the different contribution of each story tothe energy dissipation capacity of a frame, 𝑊 is the totalweight of the structure, and, finally,𝐶

𝑦and𝐷

𝑦are the seismic

coefficient and displacement at first yield, which are obtainedthrough pushover analysis.

From extensive statistical studies, 𝐹EH𝑖 can be estimatedas [5]

𝐹EH = min (𝐹∗EH, 1) , (8)

where

𝐹∗

EH =1

(−0.0675𝜇 + 2.82) ℎ/𝐻

× exp{−12[(ln (ℎ/𝐻) − ln (0.031𝜇 + 0.3461))

0.06𝜇 + 0.39]

2

} .

(9)

Equation (7) shows the role of the cumulative plasticrotation capacity of the structural members in the totalenergy dissipation capacity of a frame. Based on the resultsof several experimental tests of steel members collected by[10], Bojorquez et al. [11] found that the cumulative plasticrotation capacity of steel members is well represented by alognormal probability density function with a median valueequal to 0.23. Results illustrated below will show that thereis no influence of the uncertainty in the selection of thecumulative plastic rotation capacity.

The Scientific World Journal 3

Moment

Rotation

𝜃p2

𝜃pi

𝜃p1𝜃p3

𝜃pa =N

∑i=1

Figure 1: Estimation of cumulative plastic rotation.

3. Maximum Interstory Drift Index

The demand hazard curves in terms of maximum interstorydrift and the energy-based damage index introduced hereincannot be compared directly unless the maximum interstorydrift is normalized by its respective structural capacity. In thismanner, a normalized damage measure in terms of interstorydrift (denoted interstory drift damage index) needs to beformulated as follows:

𝐼𝐷𝛾=𝛾𝐷

𝛾𝑢

, (10)

where 𝐼𝐷𝛾

characterizes damage in terms of maximum inter-story drift; and 𝛾

𝐷and 𝛾𝑢represent the demand and capacity

of the structure, respectively, in these terms. From pushoveranalyses of the steel frames under consideration, 𝛾

𝑢was found

to be close to 0.05 [11]. Note that while the energy damageindex takes into account explicitly the cumulative demandsdue to plastic deformation, 𝐼

𝐷𝛾is based on peaks demands

in such a way that the effect of ground motion duration isnot explicitly considered. This can be better explained usingthe definition of cumulative plastic rotation which is one ofthe most important parameters to estimate the energy-baseddamage index and it is schematically illustrated in Figure 1.

4. Steel Moment Resisting Frames andEarthquake Ground Motion Records

4.1. Steel Moment Resisting Frames. Six moment resistingsteel frames having 4, 6, 8, 10, 14, and 18 stories wereconsidered for the studies reported herein. The frames aredenoted as F4, F6, F8, F10, F14, and F18, respectively. Asshown in Figure 2, the frames (designed according to theMexico City Building Code, MCBC) have three- to eight-meter bays and interstory heights of 3.5 meters. Each framewas provided with ductile detailing and its lateral strengthwas established according to the MCBC. A36 steel was usedfor the beams and columns of the frames. An elastoplasticmodel with 3% strain-hardening was used tomodel the cyclicbehavior of the steel elements. As discussed by Bojorquez

Table 1: Relevant characteristics of the moment resisting steelframes.

Frame Number of stories 𝑇1(s) 𝐶

𝑦𝐷𝑦(m)

F4 4 0.90 0.45 0.136F6 6 1.07 0.42 0.174F8 8 1.20 0.38 0.192F10 10 1.37 0.36 0.226F14 14 1.91 0.25 0.30F18 18 2.53 0.185 0.41

Variable number of stories

8.0m 8.0m 8.0m

3.5m

3.5m

3.5m

3.5m

3.5m

3.5m

3.5m

3.5m

Figure 2: Geometrical characteristic of the MDOF steel frames.

and Rivera [12], this model provides a good approximation tothe actual hysteretic behavior of steel members. The columnsin the first story were modeled as clamped at their bases.Second order effects were explicitly considered, and 3% ofcritical damping was used for the two first modes of theframes during the nonlinear dynamic analyses. Relevantcharacteristics for each frame are summarized in Table 1. Inthis table, 𝑇

1denotes fundamental period of vibration, 𝐶

𝑦

denotes seismic coefficient at yield, and 𝐷𝑦denotes roof

displacement at yield.

4.2. Seismic Records. A set of 23 narrow-band groundmotions recorded at Lake Zone sites of Mexico City wasconsidered. Particularly, all motions were recorded at siteshaving soil periods of two seconds during seismic eventswith magnitudes of seven or larger and having epicenterslocated at distances of 300 km or more from Mexico City.Some important characteristics of the records are summa-rized in Table 2. While PGA and PGV denote the peakground acceleration and velocity, respectively, the durationwas estimated according to Trifunac and Brady [13]. Theseismic records under consideration were established byrotating both horizontal components of a motion recorded ata given station so that its Arias intensity [14] was maximized.


Table 2: Seismic records.

Record Date Magnitude Station PGA (cm/s2) PGV (cm/s) Duration (s)1 25/04/1989 6.9 Alameda 45.0 15.6 45.342 25/04/1989 6.9 Garibaldi 68.0 21.5 733 25/04/1989 6.9 SCT 44.9 12.8 65.734 25/04/1989 6.9 Sector Popular 45.1 15.3 79.135 25/04/1989 6.9 Tlatelolco TL08 52.9 17.3 56.556 25/04/1989 6.9 Tlatelolco TL55 49.5 17.3 49.917 14/09/1995 7.3 Alameda 39.3 12.2 53.68 14/09/1995 7.3 Garibaldi 39.1 10.6 86.89 14/09/1995 7.3 Liconsa 30.1 9.62 50.9410 14/09/1995 7.3 Plutarco Elıas Calles 33.5 9.37 77.5711 14/09/1995 7.3 Sector Popular 34.3 12.5 100.7612 14/09/1995 7.3 Tlatelolco TL08 27.5 7.8 85.7613 09/10/1995 7.5 Cibeles 14.4 4.6 83.0614 09/10/1995 7.5 Cordoba 24.9 8.6 94.1015 09/10/1995 7.5 Liverpool 17.6 6.3 104.9516 09/10/1995 7.5 Plutarco Elıas Calles 19.2 7.9 104.4417 11/01/1997 6.9 CU Juarez 16.2 5.9 62.0918 11/01/1997 6.9 Centro urbano Presidente Juarez 16.3 5.5 60.7119 11/01/1997 6.9 Garcıa Campillo 18.7 6.9 84.8920 11/01/1997 6.9 Plutarco Elıas Calles 22.2 8.6 56.3421 11/01/1997 6.9 Est. # 10 Roma A 21.0 7.76 76.0922 11/01/1997 6.9 Est. # 11 Roma B 20.4 7.1 74.0623 11/01/1997 6.9 Tlatelolco TL55 13.4 6.5 55.37

The narrow-band records exhibit similar values of parameter𝑁𝑝, which is an indicator of the characteristics of their

spectral shape [15]. As a result, there is a strong similaritybetween their spectral shapes.This is illustrated in the log-logplot included in Figure 3, which shows response spectra of allrecords scaled to the same spectral acceleration for a periodof 1.2 sec (fundamental period of vibration of frame F8). Thesimilitude exhibited by all spectra indicates that the spectralacceleration is a good indicator of the damage potential of theground motions and emphasizes the good correspondencethat exists between parameter𝑁

𝑝and the spectral shape.

5. Seismic Vulnerability Assessment

One of the main objectives of earthquake engineering is toquantify, through the consideration of all possible earthquakeground motion intensities at a site, the seismic reliabilityimplicit in structures. Probabilistic seismic demand analy-sis (PSDA) is used as a tool for estimating the reliabilityof structures through the evaluation of the mean annualfrequency of exceeding a specified value of an earthquakedemand parameter EDP (e.g., interstory drift, normalizedplastic hysteretic energy, etc.). Based on past studies [16,17] and considering the total probability theorem, a proba-bilistic seismic demand analysis can be carried out through

1

10

100

1000

0.1 1 10T (s)

Sa (c

m/s2)

Figure 3: Elastic response spectra for records scaled up to the samevalue of Sa(𝑇

1) for 𝑇

1of 1.2 sec (3% of critical damping).

the consideration of the mean annual rate of exceeding agiven value of EDP:

𝜆EDP (𝑥)

= ∑ ]𝑖∫

IM∫

𝑀

∫

𝑅

𝑃 [EDP > 𝑥 | IM,𝑀, 𝑅]

× 𝑓 (IM | 𝑀, 𝑅) 𝑓 (𝑀, 𝑅) 𝑑𝑟 𝑑𝑚𝑑 (im) ,(11)


where 𝜆EDP(𝑥) is the mean annual frequency of EDP exceed-ing the value 𝑥, ]

𝑖is the rate of earthquakes for source 𝑖,

𝑓(IM | 𝑀, 𝑅) is the conditional distribution function ofthe intensity measure (IM) given values of magnitude (𝑀)and distance (𝑅), 𝑓(𝑀, 𝑅) is the joint probability densityfunction of 𝑀 and 𝑅, and, finally, 𝑃[EDP > 𝑥 | IM,𝑀, 𝑅]is the probability of EDP exceeding 𝑥 given IM, 𝑀, and 𝑅(if 𝑥 corresponds to the capacity of the structure, this termrepresents the fragility curves of the system). If 𝑃[EDP >

𝑥 | IM,𝑀, 𝑅] = 𝑃[EDP > 𝑥 | IM], then the IM is said tobe sufficient [18, 19] since its ability to predict the structuralresponse is independent of 𝑀 and 𝑅, given IM. It has beenshown that thespectral acceleration at first mode of vibrationSa(𝑇1) is sufficient with respect to magnitude and distance

[19]. As suggested before, the records used herein allow theuse of a scaling criteria based on Sa(𝑇

1): (a) first, due to

sufficiency of Sa(𝑇1) with respect to 𝑀 and 𝑅, (b) second,

due to the similar spectral shape of the records, and (c) third,because no bias in nonlinear structural response is observedfor different scaling levels of the records under consideration[15]. Within this context, (11) can be expressed as

𝜆EDP (𝑥) = ∫Sa(𝑇1)

𝑃 [EDP > 𝑥 | Sa (𝑇1) = sa] 𝑑𝜆Sa(𝑇1) (sa) ,

(12)

where 𝑑𝜆Sa(𝑇1)(sa) = 𝜆Sa(𝑇1)(sa) − 𝜆Sa(𝑇1)(sa + 𝑑sa) is thedifferential of the ground motion hazard curve expressedin terms of Sa(𝑇

1). Equation (12) was used to evaluate the

structural reliability of the steel frames in terms of two EDPs:interstory drift index and energy damage index. A lognormaldistribution is considered to evaluate 𝑃[EDP > 𝑥 | Sa].

6. Incremental Dynamic Analysis andFragility Curves

The first step to assess the seismic vulnerability is theestimation of structural demands in terms of both damagemeasures under consideration at different intensity levelsof the spectral acceleration (incremental dynamic analysis).For the sake of brevity only results for the energy-baseddamage index are discussed and incorporated into the papersince for this damage measure it is important to observe theinfluence of the uncertainty in the use of 𝜃

𝑝𝑎. For this aim,

the structural damage in the frames under consideration wascomputed to assess the effect of explicitly considering theuncertainty of 𝜃

𝑝𝑎of the structural members. A lognormal

probability density function with a median value of 0.23was used to describe the variation of the cumulative plasticrotation capacity at the ends of the beams. For illustrativepurposes, four standard deviations of the natural logarithmwere considered: 0, 0.1, 0.3, and 0.5. A standard deviation ofzero corresponds to the mean values. While it was assumedthat the value of 𝜃

𝑝𝑎varied in height according to the

lognormal density function, the value of 𝜃𝑝𝑎

for all beamswithin a story was considered equal. The influence of theuncertainty of 𝜃

𝑝𝑎is illustrated through the results obtained

from incremental dynamic analyses of all frames underconsideration. For this purpose, the frames were subjected

to the ground motions included in Table 2 scaled to achievethe same spectral ordinate at the period corresponding tothe first mode of vibration of each particular frame. A widerange of motion intensities were considered for this purpose.Figure 4 compares the median values of 𝐼DEN for the steelframes. The horizontal axis considers the different intensitylevels quantified through the spectral acceleration associatedto the first mode of vibration. The comparison suggests thatthere is no significant influence of the level of uncertaintyof 𝜃𝑝𝑎

in the damage estimates for the frames. In general,the damage estimates corresponding to the different levels ofuncertainty are quite similar for each particular frame. It canbe concluded that it is reasonable to estimate the structuraldamage through the consideration of median cumulativerotation capacities and moreover to compute 𝑃[EDP > 𝑥 |

Sa] for the damage measure under consideration.In Figure 5 the damage distribution of frame F10 in terms

of 𝐼DEN is illustrated. It can be observed that the level ofdamage in all the beams within a story is practically the same,which is very important to represent the hysteretic energycapacity in a steel frame through (7). As it was discussedbefore, a value of 𝐼DEN equal to one theoretically impliesstructural failure at the critical stories of the frame. Note thataccording to the figure, the beams located from the secondto the fifth story have the largest level of damage which isequal to one. For this reason, it can be concluded that 𝐼DENis a useful global parameter that correlates very well with thelevel of damage in the critical stories of steel frames.

The second step before computing the mean annual rateof exceeding the structural demands is the estimation offragility curves. By considering a lognormal distribution toassess 𝑃[EDP > 𝑥 | Sa], the probability that EDP exceeds 𝑥given Sa(𝑇

1) is given by

𝑃 (EDP > 𝑥 | Sa (𝑇1) = 𝑠𝑎)

= 1 − Φ(

ln𝑥 − 𝜇lnEDP|Sa(𝑇1)=𝑠𝑎��lnEDP|Sa(𝑇1)=𝑠𝑎

) .

(13)

In (13), 𝜇lnEDP|Sa(𝑇1)=𝑠𝑎 and ��lnEDP|Sa(𝑇1)=𝑠𝑎 are the samplemean and standard deviation for the EDP, respectively, andΦ(⋅) is the standard normal cumulative distribution function.Although the maximum interstory drift has been foundto be well represented by a lognormal distribution [19], aKolmogorov-Smirnov test was developed to validate the useof this distribution for the case of the normalized plastichysteretic energy, 𝐸

𝑁, defined as

𝐸𝑁=

𝐸𝐻

𝐶𝑦𝐷𝑦𝑊, (14)

where 𝐸𝐻is the total plastic hysteretic energy dissipated by

the structure during the ground motion, 𝐷𝑦and 𝐶

𝑦are the

global displacement and seismic coefficient at first yield, and𝑊 is the total weight of the structure. Note that the productof the seismic coefficient times𝑊 is equal to the base shearof the structure at first yield. Figure 6 compares the actual𝐸𝑁probability density function corresponding to frame F10

and Sa(𝑇1) = 1200 cm/sec2, with its analytically derived


0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

Sa (cm/s2)

I DEN

(a) Frame F4

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000Sa (cm/s2)

I DEN

(b) Frame F6

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

Sa (cm/s2)

I DEN

(c) Frame F8

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

Sa (cm/s2)

I DEN

(d) Frame F10

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

MedianStd 0.1

Std 0.3Std 0.5

Sa (cm/s2)

I DEN

(e) Frame F14

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

MedianStd 0.1

Std 0.3Std 0.5

Sa (cm/s2)

I DEN

(f) Frame F18

Figure 4: Estimation of 𝐼DEN at different intensity levels for the frames considering uncertainties in the value of 𝜃𝑝𝑎.


≤ 1.000.80 ≤

< 0.800.60 ≤< 0.600.40 ≤

< 0.400.20 ≤

< 0.200.10 ≤

< 0.10

Figure 5: Damage distribution in frame F10, 𝐼DEN = 1.0.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20 40 60 80

Lognormal

f(E

N)

EN

Figure 6: Probability density function for 𝐸𝑁

demands of steelframe F10 at a spectral acceleration equal to 1200 cm/s2.

lognormal density function.The K-S test for the data set sug-gests that 𝐸

𝑁is well represented by the lognormal probability

density function. Similar results have been obtained for otherframes and a wide range of values of spectral acceleration.For this reason fragility curves are obtained by consideringa lognormal probability density function for both damagemeasures here selected.

7. Cyclic Interstory Drift Capacity for SteelMoment Resisting Frames

In this study, structural reliabilities were established throughthe use of (12) and the ground motion seismic hazard

0.0001

0.001

0.01

0.1

1

10

1 10 100 1000

Mea

n an

nual

rate

of

exce

edan

ce

Sa (cm/s2)

Figure 7: Seismic hazard curves under consideration.

0.00001

0.0001

0.001

0.01

0.1

1

0.1 1 10Measure of damage

ID (normalized dissipated hysteretic energy)

Failure of the system

ID (maximum interstory dri)

Mea

n an

nual

rate

of

exce

edan

ce

Figure 8: Hazard curves in terms of 𝐼DEN and 𝐼𝐷𝛾

for frame F4.

curves corresponding to the Secretaria de Comunicacionesy Transportes (SCT) site in Mexico City and established byAlamilla [20], which are illustrated in Figure 7 for all theperiods of the selected structural steel frames. It can beobserved that the curve with the largest mean annual rate ofexceeding a spectral acceleration is the blue line which wasobtained for a period close to 2 seconds (the structural periodof the frame F14). This result was obtained because the soil atthe site exhibits a period of two seconds, which is consideredrepresentative of the sites where the ground motions underconsideration were recorded.

A 𝜃𝑝𝑎

= 0.23 was used to characterize the normalizedplastic hysteretic energy capacity of the beams as it was pre-viously discussed. Figure 8 summarizes the demand hazardcurves in terms of 𝐼

𝐷𝛾and 𝐼DEN for frame F4 (values larger

than one were plotted for illustrative purposes). Three zonescan be appreciated in the figures. The first one correspondsto small values of 𝐼

𝐷(blue box), which would commonly be


associated to the serviceability limit state. In this range of 𝐼𝐷,

the mean annual rate of exceedance is larger for 𝐼𝐷𝛾

thanfor 𝐼DEN. This seems logical since the level of displacementcontrol required by serviceability implies minimum or noplastic demands in the structural elements. A second zone,corresponding to intermediate values of 𝐼

𝐷(close to 0.5),

can be noticed in Figure 8 (orange box). In this zone, bothdemand hazard curves exhibit similar ordinates, implyingthat design for intermediate levels of damage is not sensibleto the measure of damage used to guarantee an adequateperformance of the frame. Finally, a third zone (red box) canbe appreciated for values of 𝐼

𝐷close to one. Because this zone

relates to failure, it is usually deemed as the most importantin terms of practical seismic design. The results suggest thatunder some circumstances, an unsatisfactory design can beobtained if measures of the ground motion duration or ofcumulative demands are not taken into account explicitly.This discussion is valid for all the frames under consideration.

Because in most seismic design codes the principalparameter to promote adequate structural performance isthe maximum interstory drift index, it is important to offermaximum interstory drift thresholds that consider the effectof cyclic cumulative deformation (denoted cyclic interstorydrift capacity, 𝛾CC). These thresholds should be establishedin such manner that the structural reliabilities associatedto the failure of the frames are similar in terms of bothdamage indices used herein. Table 3 summarizes the reducedinterstory drift thresholds that account for energy demands.Note that the maximum reduction in terms of drift thresholdoccurs for the frames whose fundamental period of vibrationis close to two seconds (period of the soil at the site). TechnicalRequirements for Seismic Design of the MCBC considers athreshold of 0.03 for seismic design of ductile steel frames.This threshold is conservative if compared to the 0.05 valueestimated for the frames under consideration. Nevertheless,the cyclic interstory drift ratio thresholds included in Table 3indicate that the 0.03 threshold may not be conservativeenough for structures located in the Lake Zone, particularlyfor those whose period of vibration is similar to that of thesoil. In the latter case, a threshold or cyclic drift capacity of0.02 would seem more appropriate. To better understand thepoint of view of the authors notice that the annual rate ofexceeding the energy-based damage index equal to 1 is about0.0002. Hence, the damage index for maximum interstorydrift at an annual rate of exceedance of 0.0002 is about0.65. Since the damage index for maximum interstory driftis defined as the drift demand divided by 0.05, the cyclicinterstory drift capacity for F4 is computed using the productof 0.65 and 0.05 (see Table 3).

8. Toward Seismic Design of SteelFrames Structures Using an Energy-BasedDamage Index by means of the CyclicInterstory Drift

Because it provides a simple manner to estimate the max-imum seismic demands on earthquake-resistant structures,one of the most important tools for seismic design is a well

Table 3: Cyclic interstory drift capacity.

Frame 𝑇1(s) 𝛾CC

F4 0.90 0.032F6 1.07 0.029F8 1.20 0.026F10 1.37 0.023F14 1.91 0.018F18 2.53 0.023

formulated seismic design spectrum. Nevertheless, severalissues should be taken into consideration when formulatingappropriate design spectra within a code format. A firstissue that will be discussed herein is the inclusion of theeffect of cumulative deformation demands during the seismicdesign of an earthquake-resistant structure. In this paper,cumulative demands are explicitly contemplated through theformulation of a damage index that makes explicit con-sideration of the cumulative plastic deformation demands.It was observed that the use of this energy-based damageindex within a design context results in reasonable estimatesof structural damage in regular steel frames. Within thiscontext, energy spectra can be used for structural evaluationof complex structures and thus constitute a relevant tool forstructural assessment. Note that the energy-based evaluationwas formulated in terms of the energy dissipation capacityof earthquake-resistant structures, in such a way that oncethis capacity is established by using information providedfrom experimental testing of steel elements, the only otherinformation required in the form of an energy demand canbe derived from normalized hysteretic energy spectra.

A second issue that will be addressed is the inclusionof specific reliability levels associated to the structures to bedesigned. Regarding this, Bojorquez et al. [5] proposed anearthquake-resistant evaluation procedure for steel framesthat considers the use of normalized hysteretic energy spectrawith uniform annual failure rates. The combination of theenergy-based evaluation procedure developed in this paperwith the formulation of energy spectra with uniform annualfailure rates constitutes itself an excellent alternative toincorporate cumulative demands within specific structuralreliability settings.

Finally, this study shows a simplified way for taking intoaccount cumulative damage in steel structures by means ofthe cyclic interstory drift. The application of this parameteris easier than the use of the energy approach, which is analternative for a seismic design of steel framed structuresthat considers cumulative demands. For the case of struc-tures subjected to long duration ground motion and havingsimilar characteristics as those under consideration herein, areduction of about 30% in the maximum allowable interstorydrift is recommended to incorporate the effect of cumulativeplastic deformation demands.

9. Conclusions

An energy-based damage model for multidegree-of-freedomsteel framed structures that accounts explicitly for the effects


of cumulative plastic deformation was used to estimateimproved thresholds for the maximum interstory drift. Theevaluation of the reduced drift limit, denoted cyclic interstorydrift capacity, was based on the use of the energy damageindex and was targeted to achieve similar levels of structuralreliability at failure of the frames in terms of interstory driftand energy.

For the serviceability limit state, seismic design is con-trolled by maximum interstory drift demands. Nevertheless,in terms of failure, the results presented herein suggest thatunder some circumstances an unsatisfactory design can beobtained if measures of the ground motion duration or ofcumulative demands are not taken into account explicitly.This is particularly important for structures located in verysoft soils and having a fundamental period of vibration closeto the dominant period of the soil.

Interstory drift ratio thresholds currently used to pro-mote adequate structural performance during severe groundmotions usually yield conservative seismic design. Never-theless, these thresholds need to be carefully assessed forthe seismic design of structures located at sites capable ofgenerating long duration motions. Particularly, a cyclic driftcapacity of 0.02 seems a better value to achieve adequatestructural performance of ductile steel structures located inthe Lake Zone of Mexico City, compared to the current valueof 0.03 formulated in current Mexican building codes, whichindicates a reduction of about 30% of the actual maximuminterstory drift capacity.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The support given by El Consejo Nacional de Ciencia yTecnologıa to the first author under Grant CB-2011-01-167419 is appreciated. Financial support also was receivedfrom the Universidad Autonoma de Sinaloa under GrantPROFAPI 2013/025, the Universidad Nacional Autonoma deMexico under Project PAPIIT-IN102114, and the UniversidadAutonoma Metropolitana.

References

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Research Article Estimation of Cyclic Interstory Drift